CN113110048A - Nonlinear system output feedback adaptive control system and method adopting HOSM observer - Google Patents

Nonlinear system output feedback adaptive control system and method adopting HOSM observer Download PDF

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CN113110048A
CN113110048A CN202110392290.9A CN202110392290A CN113110048A CN 113110048 A CN113110048 A CN 113110048A CN 202110392290 A CN202110392290 A CN 202110392290A CN 113110048 A CN113110048 A CN 113110048A
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CN113110048B (en
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祝汝松
林辰龙
张双喜
王生利
康金峰
王鹏飞
杜立强
兰其龙
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Equipment Design and Testing Technology Research Institute of China Aerodynamics Research and Development Center
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Abstract

The embodiment of the application provides a nonlinear system output feedback adaptive control system and method adopting an HOSM (hyper extended synchronous compensator) observer, and relates to the technical field of adaptive control. Aiming at a single-input single-output uncertain nonlinear system with actuator limitation, particularly under the conditions of control gain uncertainty and actuator speed limitation, the characteristic of an HOSM observer is utilized, and the unmeasured state of the system can be accurately estimated under the condition of only needing the output quantity of the observed system. The design of the controller can be easier due to the design of the relatively independent observer, and when the method is applied to the roll dynamic control of the delta wing aircraft, the self-adaptive adjusting capacity of the delta wing aircraft to the external environment and the robustness of a self-adaptive control system are improved.

Description

Nonlinear system output feedback adaptive control system and method adopting HOSM observer
Technical Field
The application relates to the technical field of adaptive control, in particular to a nonlinear system output feedback adaptive control system and method adopting an HOSM observer.
Background
Currently, there have been many studies on adaptive output feedback control with input constraint systems. In general, these methods require the construction of an observer to estimate the non-measurable variables in the system, and then design adaptive controllers using the observer to address these difficulties in actual system control, in conjunction with methods that have been validated to handle input constraints. However, it is very difficult to design a good observer for a nonlinear system with uncertainty, and the method proposed in the prior art is complex in design and not easy to implement, and has the problem that adaptive adjustment is not accurate enough.
For example, for controlling the dynamics of a delta wing aircraft under a large attack angle condition, the currently adopted model reference adaptive control method needs all the states of a controlled system to be measurable, and is not an output feedback adaptive control method. In addition, many other output feedback adaptive control methods do not take into account the fact that the system is accompanied by uncertainty in the control gain when there is a limit on the actuator velocity. This may result in poor adaptive tuning and robustness of the system control.
Disclosure of Invention
In view of the above, embodiments of the present application provide a nonlinear system output feedback adaptive control system and method using a HOSM observer to improve the above problem.
In a first aspect, the present application provides a nonlinear system output feedback adaptive control system using an HOSM observer, including a state observer using an HOSM, a dynamic system of an object to be controlled, an adaptive law, and a control law;
a single input single output nonlinear system with input actuator limitations is proposed:
Figure BDA0003017236720000021
v＝sat(η(u))
where x (t) is the measurable output, φiIs a known non-linear smoothing function that satisfies the consistent Lipschitz condition, θiAs an unknown constant, bcFor unknown constant control gain, v is actuator output, d1 is a bounded slow time-varying disturbance, and u is the controller output signal. sat (-) and η (-) are the amplitude saturation function and the rate saturation function of the actuator, respectively, and are defined as:
Figure BDA0003017236720000022
Figure BDA0003017236720000023
wherein eta isuAnd ηlSaturating the upper and lower limits for a known amplitude,
Figure BDA0003017236720000024
and
Figure BDA0003017236720000025
saturating the upper and lower limits for a known rate;
rewriting the nonlinear system with single input and single output limited by input actuators into the form of the following equation of state:
Figure BDA0003017236720000026
Figure BDA0003017236720000027
Figure BDA0003017236720000028
Figure BDA0003017236720000031
v＝sat(η(u))
wherein the content of the first and second substances,
Figure BDA0003017236720000032
θ＝[-θ1，-θ2，...-θr，d1]T，φ＝[φ1，φ2，...，φr，1]T，x1being the only measurable output of the nonlinear system, bcFor unknown gain, v is the output of the actuator, u is the output signal of the controller, d1Is a bounded, slowly time-varying perturbation;
the state observer includes:
Figure BDA0003017236720000033
Figure BDA0003017236720000034
Figure BDA00030172367200000311
Figure BDA0003017236720000035
Figure BDA0003017236720000036
wherein the content of the first and second substances,
Figure BDA0003017236720000037
the 0 th time derivative of the time varying signal a (t),
Figure BDA0003017236720000038
is the 1 st time derivative of the time varying signal a (t),
Figure BDA0003017236720000039
is the n-1 order time derivative of the time varying signal a (t),
Figure BDA00030172367200000310
is the nth order time derivative of the time-varying signal a (t), a (t) e [0, ∞), L, λ1...λnAre the HOSM observer parameters.
In an alternative embodiment, a dynamic system associated with the actuator saturation limit is configured for actuator saturation limits for the actuator, the dynamic system being driven by an actuator saturation error, the dynamic system generating a dynamic signal ζ for use in a subsequent controller design, the dynamic system comprising:
ζ＝[ζ1，ζ2，...，ζn]T
Figure BDA0003017236720000041
Figure BDA0003017236720000042
Figure BDA0003017236720000043
wherein b isi1,2, n is an adjustable normal number,
Figure BDA0003017236720000044
is an unknown gain bcIs updated by the adaptive law, Δ u is v-u, v is actuator' sAnd the output u is an output signal of the controller, and zeta is a dynamic signal which has a correlation relation with the saturation error of the actuator.
In an alternative embodiment, the control law comprises:
Figure BDA0003017236720000045
wherein k is1> 0 is an optional normal number,
Figure BDA0003017236720000046
for a vector of unknown parameters thetaTIs estimated by the estimation of (a) a,
Figure BDA0003017236720000047
for estimation of the tracking error vector e, the tracking error vector
Figure BDA0003017236720000048
Wherein the content of the first and second substances,
Figure BDA0003017236720000049
as a conventional tracking error vector, - ζ is an extended dynamic error term caused by actuator saturation Δ u, Δ u being v-u, v being the output of the actuator, u being the output signal of the controller, B being diag ([ B)1，b2，…，bn]) Is a diagonal matrix, biN is an adjustable normal number, phi, 1,21，φ2，...，φr，1]T。
In an alternative embodiment, the estimation of the unknown parameter vector
Figure BDA00030172367200000410
And estimation of unknown gain
Figure BDA00030172367200000411
The calculation is performed according to the following adaptation law:
Figure BDA00030172367200000412
Figure BDA00030172367200000413
wherein, Γ ═ ΓT> 0 is a symmetric positive definite gain matrix, gammabA gain of a normal number > 0, σ1，σ2＞0，
Figure BDA0003017236720000051
And
Figure BDA0003017236720000052
is the e-modification of the adaptation law.
In a second aspect, the present application provides a nonlinear system output feedback adaptive control method using a HOSM observer, comprising the steps of:
s1, a single input single output nonlinear system with input actuator constraints is proposed:
Figure BDA0003017236720000053
v＝sat(η(u))
where x (t) is the measurable output, φiIs a known non-linear smoothing function that satisfies the consistent Lipschitz condition, θiAs an unknown constant, bcFor unknown constant control gain, v is actuator output, d1 is a bounded slow time-varying disturbance, and u is the controller output signal. sat (-) and η (-) are the amplitude saturation function and the rate saturation function of the actuator, respectively, and are defined as:
Figure BDA0003017236720000054
Figure BDA0003017236720000055
wherein eta isuAnd ηlSaturating the upper and lower limits for a known amplitude,
Figure BDA0003017236720000056
and
Figure BDA0003017236720000057
saturating the upper and lower limits for a known rate;
s2: the single input single output nonlinear system with input actuator limits is rewritten as the following equation of state form:
Figure BDA0003017236720000058
Figure BDA0003017236720000059
Figure BDA0003017236720000061
Figure BDA0003017236720000062
v＝sat(η(u))
wherein the content of the first and second substances,
Figure BDA0003017236720000063
θ＝[-θ1，-θ2，...-θr，d1]T，φ＝[φ1，φ2，...，φr，1]T，x1being the only measurable output of said non-linear system, bcFor unknown gain, v is actuator output, u is controller output signal, d1Is a bounded, slowly time-varying perturbation;
s3: constructing a state observer adopting HOSM, a dynamic system of an object to be controlled, a self-adaptation law and a control law, wherein the state observer comprises:
Figure BDA0003017236720000064
Figure BDA0003017236720000065
Figure BDA0003017236720000066
Figure BDA0003017236720000067
Figure BDA0003017236720000068
wherein the content of the first and second substances,
Figure BDA0003017236720000069
the 0 th time derivative of the time varying signal a (t),
Figure BDA00030172367200000610
is the 1 st time derivative of the time varying signal a (t),
Figure BDA00030172367200000611
is the n-1 order time derivative of the time varying signal a (t),
Figure BDA00030172367200000612
is the nth order time derivative of the time-varying signal a (t), a (t) e [0, ∞), L, λ1...λnAre the HOSM observer parameters.
In an alternative embodiment, a dynamic system associated with the actuator saturation limit is configured for actuator saturation limits of the actuator, the dynamic system being driven by actuator saturation errors, and generating dynamic signals for subsequent controller design, the dynamic system comprising:
ζ＝[ζ1，ζ2，...，ζn]T
Figure BDA0003017236720000071
Figure BDA0003017236720000072
Figure BDA0003017236720000073
wherein b isi1,2, n is an adjustable normal number,
Figure BDA0003017236720000074
is an unknown gain bcIs updated by the adaptive law, Δ u is v-u, v is the actuator output, u is the controller output signal, and ζ is the dynamic signal having a correlation with the actuator saturation error.
In an alternative embodiment, the control law comprises:
Figure BDA0003017236720000075
wherein k is1> 0 is an optional normal number,
Figure BDA0003017236720000076
for a vector of unknown parameters thetaTIs estimated by the estimation of (a) a,
Figure BDA0003017236720000077
for estimation of the tracking error vector e, the tracking error vector
Figure BDA0003017236720000078
Wherein the content of the first and second substances,
Figure BDA0003017236720000079
for a conventional tracking error vector, - ζ is the extended dynamic error term caused by actuator saturation Δ u, Δ u being v-u, v being the actuator output, u being the controller output signal, B being diag ([ B)1，b2，…，bn]) Is a diagonal matrix, biN is an adjustable normal number, phi, 1,21，φ2，...，φr，1]T。
In an alternative embodiment, the estimation of the unknown parameter vector
Figure BDA00030172367200000710
And estimation of unknown gain
Figure BDA00030172367200000711
The calculation is performed according to the following adaptation law:
Figure BDA0003017236720000081
Figure BDA0003017236720000082
wherein, Γ ═ ΓT> 0 is a symmetric positive definite gain matrix, gammabA gain of a normal number > 0, σ1，σ2＞0，
Figure BDA0003017236720000083
And
Figure BDA0003017236720000084
is the e-modification of the adaptation law.
The embodiment of the application provides a nonlinear system output feedback adaptive control system and method adopting an HOSM (hyper-adaptive streaming) observer, aiming at a Single Input Single Output (SISO) uncertain nonlinear system with actuator limitation, and especially under the conditions of control gain uncertainty and actuator speed limitation, the strong characteristic of the HOSM observer is utilized, and the unmeasured state of the system can be accurately estimated only under the condition of output of the observed system. The design of the controller can be easier due to the design of the relatively independent observer, and thus when the method is applied to the control of the roll dynamics of the delta wing aircraft, the self-adaptive adjusting capacity of the delta wing aircraft to the external environment and the robustness of the self-adaptive control system are improved.
In order to make the aforementioned objects, features and advantages of the present application more comprehensible, a flight control simulation example accompanied with figures is described in detail below.
Drawings
In order to more clearly illustrate the technical solutions of the embodiments of the present application, the drawings that are required to be used in the embodiments will be briefly described below, it should be understood that the following drawings only illustrate some embodiments of the present application and therefore should not be considered as limiting the scope, and for those skilled in the art, other related drawings can be obtained from the drawings without inventive effort.
Fig. 1 is a schematic structural diagram of a delta wing aircraft according to an embodiment of the present application.
Fig. 2(a) is one of schematic diagrams of a roll angle output signal and a target signal provided by an embodiment of the present application.
Fig. 2(b) is a schematic diagram of roll rate and roll rate estimation provided by an embodiment of the present application.
FIG. 2(c) is one of the control signals and actual aileron deflection diagrams provided by embodiments of the present application.
Fig. 2(d) is a schematic diagram of an estimated parameter provided in the embodiment of the present application.
Fig. 3(a) is a second schematic diagram of the roll angle output signal and the target signal provided in the embodiment of the present application.
Fig. 3(b) is a second schematic diagram of the control signal and the actual flap deflection provided by the embodiment of the present application.
Detailed Description
In order to make the objects, technical solutions and advantages of the embodiments of the present application clearer, the technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the drawings in the embodiments of the present application, and it is obvious that the described embodiments are some embodiments of the present application, but not all embodiments. The components of the embodiments of the present application, generally described and illustrated in the figures herein, can be arranged and designed in a wide variety of different configurations.
Thus, the following detailed description of the embodiments of the present application, presented in the accompanying drawings, is not intended to limit the scope of the claimed application, but is merely representative of selected embodiments of the application. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present application.
It should be noted that: like reference numbers and letters refer to like items in the following figures, and thus, once an item is defined in one figure, it need not be further defined and explained in subsequent figures.
It should be noted that the features of the embodiments of the present application may be combined with each other without conflict.
As introduced by the background, in most practical applications, the relative degree of freedom (relative degree of freedom) of the control system tends to be greater than 1, since in addition to the dynamics of the system itself, there are typically sensor and actuator dynamics, such as in flexible structures, fluid flow, combustion processes, control of aircraft, and the like. Also, in most cases, the state of the system is often not measurable except at the system output due to physical limitations or economic considerations. Uncertainties and disturbances are also common in practical devices or systems, especially unknown parameters with large uncertainties. In practical applications, it is very rare that the control system can be designed using a very accurate system model. Adaptive control methods developed over the past decades are powerful tools for resolving uncertainty in control system parameters. Moreover, many adaptive output feedback methods have been developed for use in systems where only the output is measurable.
However, when applying adaptive control methods to actual physical systems, another major difficulty, namely the constraints of the system input actuators, needs to be considered. Typically, these input constraints include amplitude and rate saturation of the actuators, such as control valve speed and travel limits in process control, control surface deflection limits in aircraft control systems, and the like. These input limitations, if not accounted for in the adaptive controller design, may result in degraded or unstable closed loop system performance.
The inventor researches to find that at present, a lot of researches are carried out on the adaptive output feedback control with an input constraint system. In general, these methods require the construction of an observer to estimate the non-measurable variables in the system, and then design adaptive controllers using the observer to address these difficulties in actual system control, in conjunction with methods that have been validated to handle input constraints. However, it is very difficult to design a good observer for a nonlinear system with uncertainty, and the method proposed in the prior art is complex in design, not easy to implement, and has the problem of inaccurate adjustment.
Further, the inventors have found that the HOSM observer can be designed independently of the dynamics of the system being controlled and is easier to design, making the adaptive control design process simpler. However, uncertainty in actuator velocity saturation control gain is not currently considered in this type of research.
For example, for the control of the dynamics of a delta wing aircraft under a large attack angle condition, a model reference adaptive control method is often adopted at present, and all states of an adaptive control system adopting the method are measurable, but the adaptive control capability to the external environment and the robustness of the adaptive control system are poor.
In view of the above, embodiments of the present application provide a system and method for output feedback adaptive control of a nonlinear system using a HOSM observer, for a Single Input Single Output (SISO) uncertain nonlinear system with actuator constraints, including those with uncertain control gain and actuator rate constraints. The principle and implementation of this scheme are explained in detail below. The embodiments described below and the keys in the embodiments can be combined with each other without conflict.
It should be noted that the above prior art solutions have drawbacks that are the results of practical and careful study by the applicant, and therefore, the discovery process of the above problems and the solutions proposed by the following embodiments of the present application for the above problems should be the contributions of the applicant to the present application in the course of the present application.
Consider a single input single output nonlinear system with input actuator constraints of the following type:
Figure BDA0003017236720000121
v＝sat(η(u))
(1)
where x (t) is the measurable output, φiIs a known non-linear smoothing function that satisfies the consistent Lipschitz condition, θiAs an unknown constant, bcControl gain for unknown constants, v actuator output, d1 is a bounded slow time varying disturbance that is an ambient uncertain disturbance and that is bounded and slowly time varying. u is the controller output signal. sat (-) and η (-) are the amplitude saturation function and the rate saturation function of the actuator, respectively, and are defined as:
Figure BDA0003017236720000122
Figure BDA0003017236720000123
wherein eta isuAnd ηlSaturating the upper and lower limits for a known amplitude,
Figure BDA0003017236720000131
and
Figure BDA0003017236720000132
the upper and lower limits are saturated for a known rate.
Rewriting the system (1) into the form of a state equation:
Figure BDA0003017236720000133
Figure BDA0003017236720000134
Figure BDA0003017236720000135
Figure BDA0003017236720000136
v＝sat(η(u))
(4)
wherein the content of the first and second substances,
Figure BDA0003017236720000137
θ＝[-θ1，-θ2，...-θr，d1]T，φ＝[φ1，φ2，...，φr，1]T，x1being the only measurable output of the nonlinear system, bcFor unknown gain, v is actuator output, u is controller output signal, d1Is a bounded, slowly time-varying perturbation;
the control objective is to design a control law using the available output x such that the output x of the system tracks a smooth reference trajectory x within a bounded error rangerAnd all other signals of the closed loop system are bounded.
It should be noted that the above equation of state (4) is a nonlinear system called Brunovsky type. Although the system covered by equation (4) appears to be limited compared to some other forms of nonlinear system, many strict feedback nonlinear systems can be converted to this standard form, and in practice many real physical systems can also be directly represented as models of this form.
Meanwhile, equation of state (4) contains uncertain control gain and control input rate constraints. The current prior art often adopts a neural network method to deal with the uncertainty. Although the neural network approach is not the focus of the discussion herein, it is also directly applicable to the approach herein. The approach herein is more practical and versatile since uncertain control gain and input rate constraints are common in practical applications.
In order to be able to effectively solve the above control problem, the system described by equation of state (4) is assumed as follows:
assumption 1. the states of equation of state (4) are only in the closed compact set ΩXIn a variation of, i.e. [ x ]1，x2，...，xn]T∈ΩX∈RnAnd the system is controllable under control input constraints. The order n of the system is known.
Assumption 2. Direction or sign of high frequency control gain of Systemc) Are known.
Suppose 3. reference trajectory xrAnd its pre-p order derivative xr (ρ)(ρ ≦ n) is known and bounded.
For a system with control input constraints, it is reasonable to assume 1. In a sense, this can also be understood as the system being input-output stable, even though this feature is hard to see from equation of state (4) itself. Without this assumption, it is not practical to expect a control input to be able to control the system to a stable range when the state of the system has an arbitrarily large value and is outside the effectively controllable range for a system with constraints on the control input.
With the above problem explanation and assumption, the embodiment of the present application provides a nonlinear system output feedback adaptive control system using the HOSM observer, and the construction process and principle of this system are explained below.
Because only the output x is in the system (4)1Are measurable and therefore require the estimation of other states in the system by designing a suitable observer. The higher order sliding mode observer (HOSM observer) has some powerful characteristics that can be used for the estimation of the unmeasured states in (4).
For a time-varying signal a (t) defined as [0, ∞ ], it consists of a bounded, unknown but Lebesgue-measurable interference signal and an unknown base signal a0(t) composition. a is0The n-th derivative of (t) has a defined Lipschitz constant L0Is greater than 0. The HOSM observer of the following form can estimate the time derivatives of the various orders of a (t).
Figure BDA0003017236720000151
Figure BDA0003017236720000152
Figure BDA00030172367200001510
Figure BDA0003017236720000153
Figure BDA0003017236720000154
Wherein the content of the first and second substances,
Figure BDA0003017236720000155
the 0 th time derivative of the time varying signal a (t),
Figure BDA0003017236720000156
is the 1 st time derivative of the time varying signal a (t),
Figure BDA0003017236720000157
is the n-1 order time derivative of the time varying signal a (t),
Figure BDA0003017236720000158
is the nth order time derivative of the time-varying signal a (t), a (t) e [0, ∞), L > 0 is a design constant, λiIs usually chosen to be λ0＝1.1，λ1＝1.5，λ2＝3，λ3＝5，λ4＝8，λ512. The main characteristics of the state observer (5) are as in lemma 1:
1) in the absence of input noise, the parameters L and λ in equation (5) are appropriately selectediAnd a is a(n)(T) if a Lipschitz constant exists, then there exists a time T, for any T > T,
Figure BDA0003017236720000159
in this characteristic, the design constant L in the formula (5) must be sufficiently large as to be larger than the Lipschitz constant L0。
2) If the input noise satisfies the inequality | a (t) -a0(t) | < ε, then after a certain time, there is a normal number μi、vi(depending only on the parameters of equation (5)), the following inequality holds:
Figure BDA0003017236720000161
Figure BDA0003017236720000162
the HOSM observer in equation (5) can overcome some of the deficiencies of the standard sliding mode, such as signal buffeting and relative degree of freedom constraints. Characteristic 1Indicating that a (t) and a can be obtained after a bounded time(i)(t) accurate estimation
Figure BDA0003017236720000163
And
Figure BDA0003017236720000164
this means that at any time T > T, the observation error is
Figure BDA0003017236720000165
Or < deltai(δ is a small positive constant). This characteristic is generally satisfied
Figure BDA0003017236720000166
Or < deltaiThe high gain observer of (2) has advantages.
The state observer using the HOSM observer can be designed independently of the system being estimated, only requiring the output of the system. Because of this feature, the controller and observer can be designed separately and the design of the controller is made easier as all states of the system can be measured. As described in and under assumption 1, the implementation of equation (5) requires an input signal a (t), i.e., x in system (4)iIs bounded, thereby ensuring the convergence of the observer. In practice, most physical systems operate in a limited interval and the rate of change of the signal is limited, so this constraint can be easily met.
Replacing a (t) in (5) with the system output x in (4)1Then (5) is a bounded observer of (4) with a bounded convergence time.
Before the control law is designed, in order to take into account the influence of the actuator saturation limit, a dynamic system is first constructed to generate a dynamic signal ζ [ ζ ] associated with the actuator saturation error1，ζ2，...，ζn]T。
Figure BDA0003017236720000171
Figure BDA0003017236720000172
Figure BDA0003017236720000173
Wherein b isi1,2, n is an adjustable normal number,
Figure BDA0003017236720000174
is b in (4)cIt is updated by the following adaptation law (15). Δ u is v-u, v is the output of the actuator, u is the output signal of the controller, and ζ is a dynamic signal having a correlation with the actuator saturation error.
V is an output of the actuator, and may be an actual output of the controller. It is assumed here that v is available. If it is not measurable or accurate in a real system, the value of v can be obtained by numerical calculation using known upper and lower amplitude and rate saturation limits, as defined by (2) and (3).
In (8), an unknown gain b is includedcIs estimated by
Figure BDA0003017236720000175
Δ u contains actuator errors due to actuator velocity saturation or other constraints. As such, this can be employed to improve the problem of inaccurate adaptive adjustment.
Further, the tracking error vector after being augmented by ζ is defined as:
Figure BDA0003017236720000176
wherein the content of the first and second substances,
Figure BDA0003017236720000177
in the step (9), the first step is carried out,
Figure BDA0003017236720000178
Figure BDA0003017236720000179
being a conventional tracking error vector, - ζ is the extended dynamic error term caused by actuator saturation Δ u. When there is no actuator saturation, i.e., Δ u is 0, ζ → 0, 0]TEquation (9) is a conventional tracking error vector.
Due to the fact that
Figure BDA0003017236720000181
In, only x1Are measurable, and therefore in the following designs, use is made of
Figure BDA0003017236720000182
Figure BDA0003017236720000183
To replace
Figure BDA0003017236720000184
Replacing a (t) in (5) with x1Then, then
Figure BDA0003017236720000185
Can be obtained by the formula (5). According to (9), define:
Figure BDA0003017236720000186
according to the main characteristics of the state observer (5), if the initial observation error is satisfied
Figure BDA0003017236720000187
Or < deltai0Then, then
Figure BDA0003017236720000188
0 or < deltaiT > T means
Figure BDA0003017236720000189
This means that
Figure BDA00030172367200001810
(δ is a small normal number) and thus has
Figure BDA00030172367200001811
t is greater than 0. Since e includes an immeasurable amount, it is not available in the feedback control, and thus
Figure BDA00030172367200001812
Will be used in the control law such that
Figure BDA00030172367200001813
Or < deltac(δcA small positive constant). If in this case the actuator is not saturated, i.e. Δ u is 0, this is obtained
Figure BDA00030172367200001814
Or < deltacThen the control objective can be achieved.
This is a method for solving the actuator constraint, and it is also explained how to design the controller and the observer separately by using the characteristics of the HOSM observer. Thus, estimation of the tracking error vector e
Figure BDA00030172367200001815
Will be used in the following design process.
Defining scalar tracking errors es and
Figure BDA00030172367200001816
is the error vector e and
Figure BDA00030172367200001817
linear combination of medium elements:
Figure BDA00030172367200001818
wherein pi is ═ pi1，π2，...，πn-1]TIs chosen as a constant vector such that the polynomial sn-1+πn-1sn-2+...+π1Is Hurwitz stable. Thus, it is clear that if es→0，
Figure BDA00030172367200001819
Then e | → 0,
Figure BDA0003017236720000191
defining observation errors
Figure BDA0003017236720000192
And
Figure BDA0003017236720000193
comprises the following steps:
Figure BDA0003017236720000194
and define
Figure BDA0003017236720000195
Figure BDA0003017236720000196
Wherein i 1. Due to phiiTo satisfy a smooth non-linear function of a consistent Lipschitz condition, and
Figure BDA0003017236720000197
then
Figure BDA0003017236720000198
t＞0(δφA bounded normal number).
From the above relationship, the error e can be derivedsThe dynamic equation of (a) is:
Figure BDA0003017236720000199
wherein the content of the first and second substances,
Figure BDA00030172367200001910
is the unknown gain bcEstimate of (c), B ═ diag ([ B ])1，b2，…，bn]) Is a diagonal matrix, biIs defined in (8).
According to the error esThe adaptive control law of the system is designed as follows:
Figure BDA00030172367200001911
wherein k is1> 0 is an optional normal number,
Figure BDA00030172367200001912
for a vector of unknown parameters thetaTIs estimated.
Figure BDA00030172367200001913
For estimation of the tracking error vector e, the tracking error vector
Figure BDA0003017236720000201
Wherein the content of the first and second substances,
Figure BDA0003017236720000202
as a conventional tracking error vector, - ζ is an extended dynamic error term caused by actuator saturation Δ u, Δ u being v-u, v being the output of the actuator, u being the output signal of the controller, B being diag ([ B)1，b2，…，bn]) Is a diagonal matrix, biN is an adjustable normal number, phi, 1,21，φ2，...，φr，1]T。
Figure BDA0003017236720000203
And
Figure BDA0003017236720000204
the calculation is performed according to the following adaptation law:
Figure BDA0003017236720000205
Figure BDA0003017236720000206
wherein, Γ ═ ΓT> 0 is a symmetric positive definite gain matrix, gammabA gain of a normal number > 0, σ1，σ2Is greater than 0. (15) In
Figure BDA0003017236720000207
And
Figure BDA0003017236720000208
is the e-modification of the adaptive law and is used for enhancing the robustness of the adaptive law.
The following is a stability analysis of the adaptive control system described above.
2, leading: taking into account dynamic systems
Figure BDA0003017236720000209
c, d > 0, μ (t) is a bounded smoothing function, and | μ (t) | ≦ εηThen ψ (T) is also bounded and for some finite time TψLater, d/c with | psi (T) | less than or equal to exists, and T is more than Tψ。
Thus, according to the lemma 2, in the adaptive law (16)
Figure BDA00030172367200002010
And
Figure BDA00030172367200002011
is bounded.
By substituting (14) the control law (15), the closed-loop dynamics of the error es can be obtained as follows:
Figure BDA00030172367200002012
wherein the content of the first and second substances,
Figure BDA0003017236720000211
in the above second step, equations (12) and (13) are employed.
As known to those skilled in the art, theorem 1: considering the system described by equation (4), using equation (5) as the observer, then using the adaptive control laws of (15) and (16) can be such that:
1) all signals in a closed loop system are consistently bounded;
2) the system output x can track a smooth reference track x within a limited error ranger。
The following was demonstrated:
consider the following Lyapunov candidate function:
Figure BDA0003017236720000212
deriving from (17) to obtain:
Figure BDA0003017236720000213
applying the adaptation law (16) and noting that
Figure BDA0003017236720000214
Equation (18) can be converted to:
Figure BDA0003017236720000217
it is noted that
Figure BDA0003017236720000215
And applying the Young's inequality to obtain:
Figure BDA0003017236720000216
Figure BDA0003017236720000221
according to the above guiding explanation (the system output x can track a smooth reference track x within a limited error ranger) And an
Figure BDA0003017236720000222
Thus, it is possible to provide
Figure BDA0003017236720000223
And
Figure BDA0003017236720000224
is bounded. Since v is amplitude constrained and the system is input-output stable, phiiIs a smooth non-linear function with consistent Lipschitz conditions, then
Figure BDA0003017236720000225
I phi i and
Figure BDA0003017236720000226
is also bounded. Equation (19) can be rewritten as:
Figure BDA0003017236720000227
wherein, tau0Can be increased by increasing k1To reduce the error of the second term with the observer
Figure BDA0003017236720000228
Figure BDA0003017236720000229
The value of this term is correlated and can be reduced by selecting appropriate observer parameters.
Therefore, according to the formula (20), when
Figure BDA00030172367200002210
When the temperature of the water is higher than the set temperature,
Figure BDA00030172367200002211
this demonstrates the tracking error esIs consistently bounded, so that the output x of the system can track a smooth reference trajectory x with limited errorrAnd is and
Figure BDA00030172367200002212
and
Figure BDA00030172367200002213
is also bounded. According to hypothesis 1, system state xiAnd is also bounded, then ζ in equation (8) and u in equation (15) are also bounded.
The certification is over.
It should be noted that, if the adaptive control law (16) is designed as follows:
Figure BDA0003017236720000231
Figure BDA0003017236720000232
i.e., sigma-modification to enhance the robustness of the adaptive law, it can be shown that the transient state of the tracking error es and the final error magnitude can be adjusted by adjusting the parameters of the control law and the adaptive law.
In addition, if the observer (5) is operating in an ideal state, i.e. with a view to a large number of observer positions
Figure BDA0003017236720000233
Figure BDA0003017236720000234
And using a standard conventional adaptation law,
Figure BDA0003017236720000235
Figure BDA0003017236720000236
the derivative of the previous Lyapunov function V can be shown to be
Figure BDA0003017236720000237
Second derivative of
Figure BDA0003017236720000238
Are consistently bounded. Therefore, the temperature of the molten metal is controlled,
Figure BDA0003017236720000239
are consistently continuous. Meanwhile, it can be seen that V, as a function of time, approaches a limit when t → ∞. According to the Barbalt theorem, there is
Figure BDA00030172367200002310
This means that
Figure BDA00030172367200002311
When the saturation limit of the actuator disappears at this time, this means that
Figure BDA00030172367200002312
The implementation principle of the nonlinear system output feedback adaptive control system adopting the HOSM observer is explained in detail below by using the adaptive control system to control the dynamic principle of the delta wing aircraft under the condition of a large attack angle.
Referring to fig. 1, fig. 1 is a schematic structural diagram of a delta wing aircraft according to an embodiment of the present application.
When the delta wing aircraft flies at a large angle of attack, the open loop roll dynamics of the delta wing aircraft is unstable at the origin balance point, and a limit loop (wing rolling problem) exists. This local roll angle oscillation is caused by asymmetrical unsteady aerodynamic effects acting on the delta wing. Thus, the roll dynamics of delta wing aircraft require active control.
The roll dynamics of delta wing aircraft can be adjusted by means of bilaterally symmetrical movable surfaces (ailerons) located at the rear of the aircraft wings. The left aileron moves down (positive deflection) and the right aileron moves up (negative deflection), causing the aircraft to roll in the forward direction, i.e., the right wing rotates downward. The difference in the deflection of the left and right ailerons, referred to as the "differential ailerons," is the primary control input for adjusting the roll angle of the aircraft.
The general delta wing roll dynamics model is as follows:
Figure BDA0003017236720000241
Figure BDA0003017236720000242
wherein
Figure BDA0003017236720000243
Is the roll angle; p (rad/s) is the roll rate; v (delta)c) For the actual aileron differential yaw angle (rad), which is also the actual control input, v (·) is defined in (4); deltac(rad) is the aileron differential control signal generated by the control law. (23) The actual values of the unknown constants are:
θ1＝-0.018，θ2＝0.015，θ3＝-0.062，θ4＝0.009，θ5＝0.021，θ6＝0.75，θ7＝0.01。
compared with the prior art, the formula (23) adds the amplitude and the rate saturation of the differential deflection of the aileron and the constant disturbance theta7。θ6Corresponding to the indeterminate control gain b in (4)c. Here, falseProvided with only roll angle
Figure BDA0003017236720000244
Is measurable.
Note that in equation (23), the unit of angle is radian (rad), and the unit of angular velocity is radian/second (rad/s). For convenience, in the following simulation result curves, the angle is in degrees and the angular velocity is in degrees/sec.
The control aims to ensure that the roll angle of the delta wing aircraft can follow the proper roll angle command through active control under the condition that the input limit of the actuator exists
Figure BDA00030172367200002513
Based on the dynamic model of the delta wing aircraft, the principle that the nonlinear system output feedback adaptive control system adopting the HOSM observer controls the flight of the delta wing aircraft is explained in detail as follows:
the amplitude and rate saturation parameters of the differential aileron deflection in equation (23) are set to ηu20 degrees, ηlThe temperature of the steel is-20 degrees,
Figure BDA0003017236720000251
it should be noted that these values are only an optional embodiment, and can be set according to requirements in practical use.
Roll angle command
Figure BDA0003017236720000252
Reference tracking trajectory for generating roll angle through filtering by following second-order transfer function
Figure BDA0003017236720000253
Figure BDA0003017236720000254
Wherein ω isn1, xi 0.95, for smooth definitionReference tracking function
Figure BDA0003017236720000255
And derivatives thereof
Figure BDA0003017236720000256
These values are used in the control law. Roll angle
Figure BDA0003017236720000257
By following a reference tracking function
Figure BDA0003017236720000258
To trace instructions
Figure BDA0003017236720000259
Replacing a (t) in (5) with the roll angle in formula (23)
Figure BDA00030172367200002510
(23) The estimate of state p in (b) can be provided by the HOSM observer. Then can obtain
Figure BDA00030172367200002511
Figure BDA00030172367200002512
θ＝[θ1，θ2，θ3，θ4，θ5，θ7]T，bc＝θ6。
To reduce the disturbance in the estimated states, the order of the HOSM observer is taken to be 4 (i.e., in equation (5), n is 3), which is higher than the order of the system (23), but only the first two states are used for state estimation of (23). (5) The HOSM observer parameter in (1) is selected to be L-1000, lambda3＝5，λ2＝3，λ1＝1.5，λ01.1, the observer has an initial value of
Figure BDA0003017236720000261
Figure BDA0003017236720000262
The adaptive control laws are designed according to (15) and (16). Wherein the design parameter is k1＝1，ΠT＝[4]T，B＝diag([b1，b2])，b1＝0.9，b2＝0.9，Γ＝diag([10，100，1000，100，40，10])，σ1＝0.01，γb＝400，σ2＝0.01。
(16) Adaptive parameter of
Figure BDA0003017236720000263
And
Figure BDA0003017236720000264
has an initial value of
Figure BDA0003017236720000265
Figure BDA0003017236720000266
The numerical values of these parameters indicate that little information is known about the parameters prior to roll angle closed-loop control, eliminating the assumed control efficiency
Figure BDA0003017236720000267
Outside the initial value.
Roll angle command for a series of step changes
Figure BDA0003017236720000268
The simulation results of the roll angle closed-loop control are shown in fig. 2.
It can be seen that the roll angle is at the beginning of the closed loop control even in the presence of uncertainty and actuator constraints in the system
Figure BDA0003017236720000269
The reference track can be tracked within an appropriate range
Figure BDA00030172367200002610
Thereby following the step instruction
Figure BDA00030172367200002611
The estimated roll rate from the HOSM observer is substantially the same as the true roll rate. FIG. 2(c) shows that even though the control law signal is outside the range of actuator limits, the actual differential aileron deflection is well within the amplitude and velocity limits. The parameters estimated in FIG. 2(d) perform well throughout the control process, except for θ7And controlling the gain bcIn addition, most parameters do not converge to their true values.
The roll angle dynamic model in equation (23) has the same standard form as (4). To make the problem more challenging, embodiments of the present application can also be applied in a more complex kinetic model (25) where the states are paired
Figure BDA00030172367200002612
Some imaginary direct input is added. The dynamical model (25) is still adaptively controlled using the adaptive control system designed above, without any change, as will be explained in detail below:
Figure BDA0003017236720000271
Figure BDA0003017236720000272
referring to fig. 3, fig. 3(a) is a schematic diagram of a second roll angle output parameter and a second target parameter provided by the embodiment of the present application. Fig. 3(b) is a second schematic diagram of the control signal and the actual aileron deflection signal provided by the embodiment of the present application.
As can be seen by comparing FIG. 2 with FIG. 3, the roll angle in FIG. 3
Figure BDA0003017236720000273
The output response is almost the same as in fig. 2. Control signal deltacAnd actual differential aileron deflection and mapping2 with some differences. This is a natural consequence of the adaptive control system applying different dynamical models.
Therefore, it can be seen that the adaptive output feedback control method designed above has good robustness and adaptability to system changes.
In summary, the nonlinear system output feedback adaptive control system and method using the HOSM observer provided in the embodiments of the present application utilize the strong characteristics of the HOSM observer, and can accurately estimate the unmeasured state of the system only by the output of the observed system. This relatively independent observer design can make the controller design much easier, as all states of the system can be measured. An auxiliary dynamic system driven by actuator saturation errors, with appropriate parameter selection, can effectively cope with the actuator constraints of the system. Stability analysis shows that the closed loop system is consistently bounded. The control results of the roll dynamics of the delta wing aircraft show that the proposed control method is effective and has good performance.
The above description is only for the specific embodiments of the present application, but the scope of the present application is not limited thereto, and any changes or substitutions that can be easily conceived by those skilled in the art within the technical scope of the present application should be covered within the scope of the present application. Therefore, the protection scope of the present application shall be subject to the protection scope of the claims.

Claims (8)

1. A nonlinear system output feedback self-adaptive control system adopting an HOSM observer is characterized by comprising a state observer adopting the HOSM, a dynamic system of an object to be controlled, a self-adaptive law and a control law;
a single input single output nonlinear system with input actuator limitations is proposed:
Figure FDA0003017236710000011
v＝sat(η(u))
where x (t) is the measurable output, φiIs a known non-linear smoothing function that satisfies the consistent Lipschitz condition, θiAs an unknown constant, bcControlling gain for unknown constants, v being actuator output, d1For a bounded, slowly time-varying disturbance, u is the controller output signal. sat (-) and η (-) are the amplitude saturation function and the rate saturation function of the actuator, respectively, and are defined as:
Figure FDA0003017236710000012
Figure FDA0003017236710000013
wherein eta isuAnd ηlSaturating the upper and lower limits for a known amplitude,
Figure FDA0003017236710000014
and
Figure FDA0003017236710000015
saturating the upper and lower limits for a known rate;
rewriting the nonlinear system with single input and single output limited by input actuators into the form of the following equation of state:
Figure FDA0003017236710000016
Figure FDA0003017236710000021
v＝sat(η(u))
wherein x is x1,
Figure FDA0003017236710000022
θ＝[-θ1,-θ2,…-θr,d1]T，φ＝[φ1,φ2,…,φr,1]T，x1Being the only measurable output of the nonlinear system, bcFor unknown gain, v is the output of the actuator, u is the output signal of the controller, d1Is a bounded, slowly time-varying perturbation;
the state observer includes:
Figure FDA0003017236710000023
wherein the content of the first and second substances,
Figure FDA0003017236710000024
the 0 th time derivative of the time varying signal a (t),
Figure FDA0003017236710000025
is the 1 st time derivative of the time varying signal a (t),
Figure FDA0003017236710000026
is the n-1 order time derivative of the time varying signal a (t),
Figure FDA0003017236710000027
is the n-1 time derivative of the time-varying signal a (t), a (t) e [0, ∞), L, λ1…λnAre the HOSM observer parameters.
2. The nonlinear system output feedback adaptive control system employing a HOSM observer of claim 1, wherein for actuator saturation limits of the actuator, a dynamic system related to the actuator saturation limits is constructed that is driven by actuator saturation errors, the dynamic system generating a dynamic signal ζ for use in subsequent controller design, the dynamic system comprising:
ζ＝[ζ1,ζ2,…,ζn]T
Figure FDA0003017236710000031
Figure FDA0003017236710000032
Figure FDA0003017236710000033
wherein b isiI is 1,2, …, n is an adjustable normal number,
Figure FDA0003017236710000034
is an unknown gain bcIs updated by the adaptive law, Δ u is v-u, v is the output of the actuator, u is the output signal of the controller, and ζ is a dynamic signal having a correlation with the actuator saturation error.
3. The nonlinear system output feedback adaptive control system with the HOSM observer of claim 1, wherein the control law comprises:
Figure FDA0003017236710000035
wherein k is1>0 is a selectable normal number which is a constant number,
Figure FDA0003017236710000036
for a vector of unknown parameters thetaTIs estimated by the estimation of (a) a,
Figure FDA0003017236710000037
for estimation of tracking error vector eTracking error vector
Figure FDA0003017236710000038
Wherein the content of the first and second substances,
Figure FDA0003017236710000039
as a conventional tracking error vector, - ζ is an extended dynamic error term caused by actuator saturation Δ u, Δ u being v-u, v being the output of the actuator, u being the output signal of the controller, B being diag ([ B)1,b2,…,bn]) Is a diagonal matrix, biI is 1,2, …, n is an adjustable normal number, phi is [ phi ]1,φ2,…,φr,1]T。
4. The nonlinear system output feedback adaptive control system with the HOSM observer of claim 3, wherein the estimation of the unknown parameter vector
Figure FDA00030172367100000310
And estimation of unknown gain
Figure FDA00030172367100000311
The calculation is performed according to the following adaptation law:
Figure FDA00030172367100000312
Figure FDA00030172367100000313
wherein, Γ ═ ΓT>0 is a symmetric positive definite gain matrix, gammab>0 is the normal number gain, σ1，σ2>0，
Figure FDA0003017236710000041
And
Figure FDA0003017236710000042
is the e-modification of the adaptation law.
5. A nonlinear system output feedback self-adaptive control method adopting an HOSM observer is characterized by comprising the following steps:
s1, a single input single output nonlinear system with input actuator constraints is proposed:
Figure FDA0003017236710000043
v＝sat(η(u))
where x (t) is the measurable output, φiIs a known non-linear smoothing function that satisfies the consistent Lipschitz condition, θiAs an unknown constant, bcControlling gain for unknown constants, v being actuator output, d1For a bounded, slowly time-varying disturbance, u is the controller output signal. sat (-) and η (-) are the amplitude saturation function and the rate saturation function of the actuator, respectively, and are defined as:
Figure FDA0003017236710000044
Figure FDA0003017236710000045
wherein eta isuAnd ηlSaturating the upper and lower limits for a known amplitude,
Figure FDA0003017236710000046
and
Figure FDA0003017236710000047
saturating the upper and lower limits for a known rate;
s2: rewriting the nonlinear system with single input and single output limited by input actuators into the form of the following equation of state:
Figure FDA0003017236710000048
Figure FDA0003017236710000051
v＝sat(η(u))
wherein x is x1,
Figure FDA0003017236710000052
θ＝[-θ1,-θ2,…-θr,d1]T，φ＝[φ1,φ2,…,φr,1]T，x1Being the only measurable output of the nonlinear system, bcFor unknown gain, v is the output of the actuator, u is the output signal of the controller, d1Is a bounded, slowly time-varying perturbation;
s3: constructing a state observer adopting HOSM, a dynamic system of an object to be controlled, a self-adaptation law and a control law, wherein the state observer comprises:
Figure FDA0003017236710000053
wherein the content of the first and second substances,
Figure FDA0003017236710000054
the 0 th time derivative of the time varying signal a (t),
Figure FDA0003017236710000055
is the 1 st time derivative of the time varying signal a (t),
Figure FDA0003017236710000056
is the n-1 order time derivative of the time varying signal a (t),
Figure FDA0003017236710000057
is the n-1 time derivative of the time-varying signal a (t), a (t) e [0, ∞), L, λ1…λnAre the HOSM observer parameters.
6. The nonlinear system output feedback adaptive control method using the HOSM observer of claim 5, wherein for actuator saturation limits of the actuators, a dynamic system related to the actuator saturation limits is constructed that is driven by actuator saturation errors, the dynamic system generating dynamic signals for use in subsequent controller design, the dynamic system comprising:
ζ＝[ζ1,ζ2,…,ζn]T
Figure FDA0003017236710000061
Figure FDA0003017236710000062
Figure FDA0003017236710000063
wherein b isiI is 1,2, …, n is an adjustable normal number,
Figure FDA0003017236710000064
is an unknown gain bcIs updated by the adaptive law, Δ u is v-u, v is the output of the actuator, u is the output signal of the controller, and ζ is a dynamic signal having a correlation with the actuator saturation error.
7. The nonlinear system output feedback adaptive control method with the HOSM observer of claim 5, wherein the control law comprises:
Figure FDA0003017236710000065
wherein k is1>0 is a selectable normal number which is a constant number,
Figure FDA0003017236710000066
for a vector of unknown parameters thetaTIs estimated by the estimation of (a) a,
Figure FDA0003017236710000067
for estimation of the tracking error vector e, the tracking error vector
Figure FDA0003017236710000068
Wherein the content of the first and second substances,
Figure FDA0003017236710000069
as a conventional tracking error vector, - ζ is an extended dynamic error term caused by actuator saturation Δ u, Δ u being v-u, v being the output of the actuator, u being the output signal of the controller, B being diag ([ B)1,b2,…,bn]) Is a diagonal matrix, biI is 1,2, …, n is an adjustable normal number, phi is [ phi ]1,φ2,…,φr,1]T。
8. The nonlinear system output feedback adaptive control method with the HOSM observer of claim 7, wherein the estimation of the unknown parameter vector
Figure FDA00030172367100000610
And estimation of unknown gain
Figure FDA00030172367100000611
According to the followingAnd (3) calculating an adaptive law:
Figure FDA00030172367100000612
Figure FDA00030172367100000613
wherein, Γ ═ ΓT>0 is a symmetric positive definite gain matrix, gammab>0 is the normal number gain, σ1，σ2>0，
Figure FDA0003017236710000071
And
Figure FDA0003017236710000072
is the e-modification of the adaptation law.
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王鹏飞，等: "非线性干扰观测器的高超声速飞行器鲁棒反演控制", 《火力与指挥控制》 *

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CN113885314A (en) * 2021-10-22 2022-01-04 电子科技大学 Nonlinear system tracking control method with unknown gain and interference
CN113885314B (en) * 2021-10-22 2023-05-23 电子科技大学 Nonlinear system tracking control method with unknown gain and interference
CN114063458A (en) * 2021-11-19 2022-02-18 中国矿业大学 Preset performance control method of non-triangular structure system independent of initial conditions
CN114063458B (en) * 2021-11-19 2022-05-20 中国矿业大学 Preset performance control method of non-triangular structure system independent of initial conditions
CN116430737A (en) * 2023-06-13 2023-07-14 中国空气动力研究与发展中心设备设计与测试技术研究所 Self-adaptive control method of input delay nonlinear system
CN116430737B (en) * 2023-06-13 2023-08-18 中国空气动力研究与发展中心设备设计与测试技术研究所 Self-adaptive control method of input delay nonlinear system

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